V. E. Johnson. Studying convergence of markov chain monte carlo algorithms using coupled sample paths. Journal of the American Statistical Association, 91(433):154--166, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....the state space S is linearly ordered, joint work with Motoya Machida [17] has shown this to be false for general posets. So the monotone case entails a somewhat stronger condition than monotonicity of P. 5 A backward coupling algorithm for the monotone case 5. 1 Forward coupling Valen Johnson [24] proposed testing for convergence of an MCMC algorithm using coupled sample paths. There are two underlying ideas involved. First, consider simultaneously running one copy of the specified Markov chain from each possible initial state, coalescing the various trajectories as they intersect and ....

Johnson, V. E. (1996). Studying convergence of Markov chain Monte Carlo algorithms using coupled sample paths. JASA 91 154--166.

.... are Schervish and Carlin (1992) Chan (1993) Frigessi, Hwang, Sheu and Di Stefano (1993) Tierney (1994) Meyn and Tweedie (1994) Ingrassia (1994) Roberts and Polson (1994) Athreya, Doss and Sethuraman (1996) Rosenthal (1995) Mengersen and Tweedie (1996) Roberts and Tweedie (1996) Johnson (1996), Roberts and Sahu (1997) Kira and Ji (1997) Robert (1998) and Diaconis and Salo Coste (1998) The initial observations from this burn in period are usually discarded. At this point the transition distribution, Q, used in the sampler may be changed to one which is optimized according to a ....

Johnson, V. E. (1996). Studying convergence of Markov chain Monte Carlo algorithms using coupled sample paths. J. Amer. Statist. Assoc. 91, 154-166.

....multiple paths that ultimately coalesce, and hence forget their initial conditions. Forgetting initial conditions is exactly what we want our MCMC simulations to do. Various lines of enquiry prior to Propp and Wilson were concerned with using these ideas to diagnose convergence (for example, Johnson, 1996, Asmussen et al. 1992) 1.2. Coupling from the past The leap made by Propp and Wilson was to note that coupling could be used more explicitly still, to guarantee rather than assess convergence. Informally, the idea is to conduct the simulation from all possible states at a random (but finite) ....

....can be exploited, then it will certainly make Exact sampling 15 the effort of exact sampling more attractive. There is a certain aesthetic appeal to truly exact sampling, but when these methods are not feasible, simulations could still be improved by using ideas from CFTP. For example, Johnson s (1996) idea of coupling a finite set of paths forward in time to identify burn in times might be improved if it were used to specify the initial set B GammaM in a CFTP algorithm: then the bias induced by stopping based on a coalescence time would be removed. ACKNOWLEDGEMENTS We are grateful to Laird ....

Johnson, V. E. (1996). Studying convergence of Markov chain Monte Carlo algorithms using coupled sample paths. J. Amer. Statist. Assoc. 91, 154--166.

....multiple paths that ultimately coalesce, and hence forget their initial conditions. Forgetting initial conditions is exactly what we want our MCMC simulations to do. Various lines of enquiry prior to Propp and Wilson were concerned with using these ideas to diagnose convergence (for example, Johnson, 1996, Asmussen et al. 1992) Exact sampling 3 1.2. Coupling from the past The leap made by Propp and Wilson was to note that coupling could be used more explicitly still, to guarantee rather than assess convergence. Informally, the idea is to conduct the simulation from all possible states at a ....

....exploited, then it will certainly make 16 P. J. Green and D. J. Murdoch the effort of exact sampling more attractive. There is a certain aesthetic appeal to truly exact sampling, but when these methods are not feasible, simulations could still be improved by using ideas from CFTP. For example, Johnson s (1996) idea of coupling a finite set of paths forward in time to identify burn in times might be improved if it were used to specify the initial set B GammaM in a CFTP algorithm: then the bias induced by stopping based on a coalescence time would be removed. ACKNOWLEDGEMENTS We are grateful to Laird ....

Johnson, V. E. (1996). Studying convergence of Markov chain Monte Carlo algorithms using coupled sample paths. J. Amer. Statist. Assoc. 91, 154--166.

....estimator variance, which can be more relevant than convergence rate as an optimality criterion (Grenander, 1993, p. 394) especially when the random field has relatively weak interactions so the sampler is rapidly mixing and convergence rate is no longer an issue. Indeed, simulation results of Johnson (1996) that apply in typical image analysis and spatial statistics applications of the Gibbs sampler, show that convergence to stationarity is achieved fairly quickly provided the interactions in the underlying random field are moderate. Many authors have utilized estimator variance as an optimality ....

....of a burn in of 20 sweeps followed by n = 10 sweeps used for estimation of ae r . For each entry in the tables, the estimator was calculated on the basis of these 10 sweeps for each of 1000 independent runs, and then the empirical variance over the runs was calculated. Simulation results of Johnson (1996) show that 20 sweeps provide an adequate burn in for the values of fi considered here. Table 1. fi = 0:1 Gibbs sampler Metropolis sampler r = 2 r = 3 r = 4 r = 5 r = 2 r = 3 r = 4 r = 5 ae r 0.015 0.0024 0.0004 0.00007 0.015 0.0024 0.0004 0.00007 oe 2 3.39 2.26 1.60 1.24 9.57 2.49 5.00 1.55 ....

Johnson, V. E. (1996). Studying convergence of Markov chain Monte Carlo algorithms using coupled sample paths. Journal of the American Statistical Association 91, 154--166.

....at least in part because computer programs for their implementation are available from their creators. In addition to these two, we discuss the methods of Geweke (1992) Roberts (1992, 1994) Ritter and Tanner (1992) Zellner and Min (1995) Liu, Liu, and Rubin (1992) Garren and Smith (1993) Johnson (1994), Mykland, Tiermey, and Yu (1995) Yu (1994) and Yu and Mykland (1994) Furthermore, we mention some related ideas from the operations research literature, focussing on the technique of Heidelberger and Welch (1983) 2.1 Gelman and Rubin (1992) Based on normal theory approximations to exact ....

....of their algorithm should be possible) Finally, the authors own empirical results are disappointing, involving substantial methodological and computational labor to provide plots that are very difficult to interpret (the point at which the estimates become unstable is open to question) 2. 9 Johnson (1994) Johnson (1994) uses the notion of convergence as the mixing of chains initialized from an overdispersed starting distribution, but from a nonstochastic point of view. Suppose a Gibbs sampling chain can be created from a stream of uniform random deviates u i (e.g. using the inversion method ....

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Johnson, V.E. (1994), "Studying Convergence of Markov Chain Monte Carlo Algorithms Using Coupled Sampling Paths," Technical Report, Institute for Statistics and Decision Sciences, Duke University.

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Johnson, V.E., (1996), "Studying Convergence of Markov Chain Monte Carlo Algorithms Using Coupled Sample Paths," to appear in the March issue of the Journal of the American Statistical Association.

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Johnson, V.E., (1996), "Studying Convergence of Markov Chain Monte Carlo Algorithms Using Coupled Sample Paths," to appear in the March issue of the Journal of the American Statistical Association.

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V. E. Johnson. Studying convergence of markov chain monte carlo algorithms using coupled sample paths. Journal of the American Statistical Association, 91(433):154--166, 1996.

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Johnson, V. E. (1994a). Studying convergence of Markov chain Monte Carlo algorithms using coupled sample paths. Report 94-07, ISDS, Duke University.

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